Learning Outcomes

Describe a sample an are and straightforward and compound occasions in it using traditional notationCalculate the probability of an occasion using standard notationCalculate the probability of 2 independent events using conventional notationRecognize when two events are mutually exclusiveCalculate a conditional probability using standard notation

Probability is the likelihood the a specific outcome or occasion happening. Statisticians and also actuaries usage probability to make predictions around events. One actuary that works for a vehicle insurance firm would, for example, it is in interested in just how likely a 17 year old male would be to obtain in a auto accident. They would usage data from past occasions to do predictions around future occasions using the characteristics of probabilities, then usage this details to calculation an insurance money rate.

You are watching: Is the likelihood that a given event will occur.

In this section, we will discover the an interpretation of an event, and also learn how to calculate the probability that it’s occurance. We will likewise practice using typical mathematical notation come calculate and describe different kinds the probabilities.



Basic Concepts

If you role a die, choose a card from deck of play cards, or randomly choose a person and observe your hair color, we are executing an experiment or procedure. In probability, we look at the likelihood of various outcomes.

We start with part terminology.


Events and also Outcomes

The an outcome of one experiment is dubbed an outcome.An event is any certain outcome or group of outcomes.A simple event is an event that cannot be broken down furtherThe sample space is the collection of all feasible simple events.

example

If we roll a traditional 6-sided die, describe the sample space and some an easy events.


Show Solution

The sample space is the set of all possible simple events: 1,2,3,4,5,6

Some instances of an easy events:

We role a 1We role a 5

Some link events:

We role a number bigger 보다 4We roll an even number

Basic Probability

Given the all outcomes room equally likely, we have the right to compute the probability of an event E utilizing this formula:

P(E)=frac extNumber of outcomes matching to the event E extTotal variety of equally-likely outcomes


examples

If we roll a 6-sided die, calculate

P(rolling a 1)P(rolling a number bigger 보다 4)
Show Solution

Recall the the sample room is 1,2,3,4,5,6

There is one outcome corresponding to “rolling a 1,” for this reason the probability is frac16There are two outcomes bigger 보다 a 4, therefore the probability is frac26=frac13

Probabilities are basically fractions, and also can be lessened to lower terms favor fractions.


This video describes this example and the previous one in detail.

Let’s to speak you have actually a bag with 20 cherries, 14 sweet and also 6 sour. If you choose a cherry at random, what is the probability the it will be sweet?


Show Solution

There space 20 possible cherries that can be picked, therefore the variety of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (sweet), for this reason the probability the the cherry will certainly be sweet is frac1420=frac710.There is one potential complication to this example, however. It should be assumed the the probability that picking any of the cherries is the very same as the probability the picking any type of other. This wouldn’t it is in true if (let united state imagine) the sweet cherries are smaller sized than the tart ones. (The cake cherries would pertained to hand much more readily when you sampled indigenous the bag.) allow us save in mind, therefore, that as soon as we evaluate probabilities in terms of the proportion of favorable to all potential cases, us rely greatly on the presumption of equal probability for every outcomes.


Try It

At some random moment, friend look at your clock and also note the minutes reading.

a. What is probability the minutes analysis is 15?

b. What is the probability the minutes analysis is 15 or less?


Cards

A conventional deck of 52 playing cards is composed of four suits (hearts, spades, diamonds and clubs). Spades and also clubs are black if hearts and diamonds room red. Each suit contains 13 cards, every of a different rank: an Ace (which in many games functions as both a short card and a high card), cards numbered 2 with 10, a Jack, a Queen and a King.


example

Compute the probability the randomly drawing one map from a deck and also getting an Ace.


Show Solution

There room 52 cards in the deck and also 4 Aces so P(Ace)=frac452=frac113approx 0.0769

We can also think of probabilities as percents: there is a 7.69% possibility that a randomly selected card will certainly be an Ace.

Notice that the smallest feasible probability is 0 – if there room no outcomes the correspond v the event. The largest feasible probability is 1 – if all possible outcomes correspond v the event.


This video demonstrates both this example and also the vault cherry example on the page.


Certain and Impossible events

An impossible occasion has a probability that 0.A specific event has actually a probability that 1.The probability of any kind of event must be 0le P(E)le 1

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In the course of this section, if girlfriend compute a probability and also get solution that is an adverse or greater than 1, you have made a mistake and should inspect your work.

Types the Events

Complementary Events

Now permit us study the probability that an event does not happen. Together in the ahead section, take into consideration the instance of roll a six-sided die and first compute the probability of roll a six: the price is P(six) =1/6. Now consider the probability the we do not role a six: there are 5 outcomes that room not a six, so the prize is P(not a six) = frac56. An alert that

P( extsix)+P( extnot a six)=frac16+frac56=frac66=1

This is no a coincidence. Consider a generic case with n feasible outcomes and an occasion E that coincides to m of this outcomes. Then the remaining nm outcomes exchange mail to E not happening, thus

P( extnotE)=fracn-mn=fracnn-fracmn=1-fracmn=1-P(E)



Complement of an Event

The complement of an event is the event “E doesn’t happen”

The notation arE is used for the complement of occasion E.We can compute the probability the the complement using Pleft(arE ight)=1-P(E)Notice likewise that P(E)=1-Pleft(arE ight)

example

If you pull a random map from a deck of play cards, what is the probability it is no a heart?


Show Solution

There space 13 understanding in the deck, so P( extheart)=frac1352=frac14.

The probability of not illustration a love is the complement: P( extnot heart)=1-P( extheart)=1-frac14=frac34


This instance is described in the complying with video.


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Probability of two independent events


example

Suppose we flipped a coin and rolled a die, and wanted to understand the probability of acquiring a head on the coin and also a 6 top top the die.


Show Solution

We might list all possible outcomes: H1,H2,h2,H4,H5,H6,T1,T2,T3,T4,T5,T6.

Notice there space 2cdot6=12 total outcomes. The end of these, only 1 is the preferred outcome, therefore the probability is frac112.


The prior instance contained two independent events. Gaining a details outcome from rolling a die had actually no influence on the result from flipping the coin.


Independent Events

Events A and B space independent events if the probability of event B developing is the very same whether or not event A occurs.


example

Are these events independent?

A same coin is tossed two times. The two events are (1) first toss is a head and (2) second toss is a head.The two occasions (1) “It will certainly rain tomorrow in Houston” and (2) “It will rain morning in Galveston” (a city near Houston).You attract a card from a deck, then attract a 2nd card without replacing the first.
Show Solution
The probability the a head comes up top top the second toss is 1/2 nevertheless of even if it is or not a head came up ~ above the an initial toss, for this reason these occasions are independent.These occasions are not independent because it is much more likely the it will rain in Galveston on job it rains in Houston than on days it does not.The probability that the second card being red counts on even if it is the first card is red or not, therefore these occasions are not independent.

When two occasions are independent, the probability that both arising is the product that the probabilities that the separation, personal, instance events.


P(A and also B) for independent events

If occasions A and B are independent, climate the probability of both A and B occurring is

Pleft(A ext and B ight)=Pleft(A ight)cdotPleft(B ight)

where P(A and also B) is the probability of occasions A and also B both occurring, P(A) is the probability of occasion A occurring, and also P(B) is the probability of event B occurring


If girlfriend look ago at the coin and die instance from earlier, you deserve to see exactly how the variety of outcomes of the an initial event multiplied by the number of outcomes in the 2nd event multiplied to same the total number of possible outcomes in the merged event.


example

In her drawer you have actually 10 pairs of socks, 6 the which room white, and also 7 tee shirts, 3 of which space white. If friend randomly reach in and also pull out a pair that socks and also a tee shirt, what is the probability both space white?


Show Solution

The probability of picking a white pair that socks is frac610.

The probability of choosing a white tee shirt is frac37.

The probability that both being white is frac610cdotfrac37=frac1870=frac935


Examples of joint probabilities are debated in this video.


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The previous instances looked at the probability the both events occurring. Now we will certainly look at the probability that either occasion occurring.


example

Suppose we flipped a coin and also rolled a die, and wanted to understand the probability of acquiring a head top top the coin or a 6 top top the die.


Show Solution

Here, there room still 12 feasible outcomes: H1,H2,h2,H4,H5,H6,T1,T2,T3,T4,T5,T6

By merely counting, we deserve to see the 7 of the outcomes have a head on the coin or a 6 on the dice or both – we usage or inclusively here (these 7 outcomes are H1, H2, h2, H4, H5, H6, T6), therefore the probability is frac712. How might we have discovered this native the separation, personal, instance probabilities?

As we would certainly expect, frac12 of this outcomes have a head, and also frac16 of this outcomes have actually a 6 top top the die. If we add these, frac12+frac16=frac612+frac212=frac812, which is no the correct probability. Looking in ~ the outcomes we can see why: the outcome H6 would have actually been count twice, due to the fact that it consists of both a head and also a 6; the probability the both a head and rolling a 6 is frac112.

If us subtract the end this dual count, we have the exactly probability: frac812-frac112=frac712.


P(A or B)

The probability of one of two people A or B occurring (or both) is

P(A ext or B)=P(A)+P(B)–P(A ext and B)


example

Suppose we draw one card from a standard deck. What is the probability the we obtain a Queen or a King?


Show Solution

There are 4 Queens and 4 queens in the deck, hence 8 outcomes corresponding to a Queen or King the end of 52 possible outcomes. Thus the probability of illustration a Queen or a King is:

P( extKing or Queen)=frac852

Note the in this case, there space no cards that space both a Queen and also a King, so P( extKing and Queen)=0. Making use of our probability rule, we might have said:

P( extKing or Queen)=P( extKing)+P( extQueen)-P( extKing and also Queen)=frac452+frac452-0=frac852


See more about this example and also the ahead one in the following video.


In the critical example, the occasions were mutually exclusive, so P(A or B) = P(A) + P(B).


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example

Suppose we attract one card from a traditional deck. What is the probability the we acquire a red map or a King?


Show Solution

Half the cards room red, therefore P( extred)=frac2652

There are 4 kings, for this reason P( extKing)=frac452

There are two red kings, for this reason P( extRed and also King)=frac252

We can then calculate

P( extRed or King)=P( extRed)+P( extKing)-P( extRed and King)=frac2652+frac452-frac252=frac2852


Try It

In her drawer you have actually 10 pairs of socks, 6 of which space white, and also 7 tee shirts, 3 the which are white. If you with in and randomly take a pair of socks and a tee shirt, what the probability at the very least one is white?


Example

The table listed below shows the number of survey topics who have actually received and not got a speeding ticket in the last year, and the color of their car. Uncover the probability the a randomly preferred person:

Has a red car and acquired a speeding ticketHas a red vehicle or acquired a speeding ticket.
Speeding ticketNo speeding ticketTotal
Red car15135150
Not red car45470515
Total60605665

Show Solution

We can see that 15 civilization of the 665 surveyed had both a red car and also got a speeding ticket, therefore the probability is frac15665approx0.0226.

Notice that having actually a red car and also getting a speeding ticket are not live independence events, so the probability of both the them arising is not simply the product that probabilities of every one occurring.

We could answer this question by simply including up the numbers: 15 human being with red cars and also speeding ticket + 135 through red cars however no ticket + 45 with a ticket yet no red vehicle = 195 people. So the probability is frac195665approx0.2932.

We additionally could have discovered this probability by:

P(had a red car) + P(got a speeding ticket) – P(had a red car and also got a speeding ticket)

= frac150665+frac60665-frac15665=frac195665.


This table example is detailed in the following explanatory video.


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Conditional Probability

In the previous ar we computed the probabilities of events that were independent of every other. We observed that acquiring a details outcome native rolling a die had no influence on the result from flipping a coin, also though we were computing a probability based upon doing them at the very same time.

In this section, we will consider events that are dependent on every other, dubbed conditional probabilities.


Conditional Probability

The probability the occasion B occurs, given that occasion A has actually happened, is stood for as

P(B | A)

This is check out as “the probability of B offered A


For example, if you draw a card from a deck, then the sample an are for the following card attracted has changed, due to the fact that you are now working v a deck that 51 cards. In the following instance we will display you how the computations for events like this are different from the computations we did in the critical section.


example

What is the probability that two cards drawn at arbitrarily from a deck of play cards will certainly both be aces?


Show Solution

It can seem that you might use the formula for the probability of two independent events and simply multiply frac452cdotfrac452=frac1169. This would certainly be incorrect, however, since the two events are no independent. If the an initial card attracted is one ace, climate the probability that the 2nd card is likewise an ace would certainly be lower since there would just be 3 aces left in the deck.

Once the first card favored is an ace, the probability that the second card favored is also an ace is called the conditional probability of illustration an ace. In this instance the “condition” is that the very first card is an ace. Symbolically, we write this as:

P(ace on second draw | one ace top top the an initial draw).

The upright bar “|” is review as “given,” therefore the above expression is brief for “The probability the an ace is attracted on the 2nd draw given that an ace was attracted on the first draw.” What is this probability? after ~ an ace is drawn on the an initial draw, there space 3 aces out of 51 complete cards left. This means that the conditional probability of drawing an ace after one ace has currently been drawn is frac351=frac117.

Thus, the probability that both cards gift aces is frac452cdotfrac351=frac122652=frac1221.


Since we understand the person has actually a red car, us are just considering the 150 civilization in the an initial row that the table. The those, 15 have a speeding ticket, for this reason P(ticket | red car) = frac15150=frac110=0.1Since we recognize the person has a speeding ticket, us are only considering the 60 civilization in the very first column that the table. That those, 15 have actually a red car, for this reason P(red car | ticket) = frac1560=frac14=0.25.

Notice native the last instance that P(B | A) is not equal to P(A | B).


These kinds of conditional probabilities room what insurance carriers use to determine your insurance allowance rates. Castle look at the conditional probability the you having actually accident, given your age, your car, your car color, her driving history, etc., and price your policy based on that likelihood.

View more about conditional probability in the complying with video.


Example

If you draw two cards indigenous a deck, what is the probability that you will get the Ace the Diamonds and a black card?


Show Solution

You can satisfy this condition by having case A or case B, as follows:

Case A) girlfriend can obtain the Ace that Diamonds an initial and climate a black card or

Case B) friend can gain a black color card an initial and climate the Ace that Diamonds.

Let’s calculation the probability of situation A. The probability the the an initial card is the Ace that Diamonds is frac152. The probability the the second card is black given that the very first card is the Ace that Diamonds is frac2651 since 26 that the remaining 51 cards are black. The probability is thus frac152cdotfrac2651=frac1102.

Now for situation B: the probability the the an initial card is black color is frac2652=frac12. The probability that the second card is the Ace that Diamonds provided that the first card is black is frac151. The probability of case B is as such frac12cdotfrac151=frac1102, the exact same as the probability of situation 1.

Recall that the probability of A or B is P(A) + P(B) – P(A and B). In this problem, P(A and also B) = 0 because the very first card cannot be the Ace the Diamonds and be a black color card. Therefore, the probability of instance A or case B is frac1101+frac1101=frac2101. The probability the you will acquire the Ace of Diamonds and a black card when drawing two cards native a deck is frac2101.


These two playing card scenarios are discussed further in the complying with video.

See more: Uci Beall Center For Art And Technology At Uc Irvine, Uc Irvine, Beall Center For Art And Technology


Try It


Since we know the test result was positive, we’re limited to the 75 females in the very first column, of i beg your pardon 5 were not pregnant. P(not pregnant | confident test result) = frac575approx0.067.Since we understand the mrs is no pregnant, we are limited to the 19 ladies in the 2nd row, of which 5 had a positive test. P(positive test result | no pregnant) = frac519approx0.263

The second result is what is usually dubbed a false positive: A positive result when the mrs is no actually pregnant.